[Brave News]: Remember last Feed View

[fallaciousreasoning]

Resolves brave/brave-browser#35968

Submitter Checklist:

• I confirm that no security/privacy review is needed and no other type of reviews are needed, or that I have requested them
• There is a ticket for my issue
• Used Github auto-closing keywords in the PR description above
• Wrote a good PR/commit description
• Squashed any review feedback or "fixup" commits before merge, so that history is a record of what happened in the repo, not your PR
• Added appropriate labels (QA/Yes or QA/No; release-notes/include or release-notes/exclude; OS/...) to the associated issue
• Checked the PR locally:
  • npm run test -- brave_browser_tests, npm run test -- brave_unit_tests wiki
  • npm run presubmit wiki, npm run gn_check, npm run tslint
• Ran git rebase master (if needed)

Reviewer Checklist:

• A security review is not needed, or a link to one is included in the PR description
• New files have MPL-2.0 license header
• Adequate test coverage exists to prevent regressions
• Major classes, functions and non-trivial code blocks are well-commented
• Changes in component dependencies are properly reflected in gn
• Code follows the style guide
• Test plan is specified in PR before merging

After-merge Checklist:

• The associated issue milestone is set to the smallest version that the
  changes has landed on
• All relevant documentation has been updated, for instance:
  • https://github.com/brave/brave-browser/wiki/Deviations-from-Chromium-(features-we-disable-or-remove)
  • https://github.com/brave/brave-browser/wiki/Proxy-redirected-URLs
  • https://github.com/brave/brave-browser/wiki/Fingerprinting-Protections
  • https://github.com/brave/brave-browser/wiki/Brave%E2%80%99s-Use-of-Referral-Codes
  • https://github.com/brave/brave-browser/wiki/Web-Compatibility-Exceptions-in-Brave
  • https://github.com/brave/brave-browser/wiki/QA-Guide
  • https://github.com/brave/brave-browser/wiki/P3A

Test Plan:

1. Open Brave News
2. Change to the following feed
3. Restart
4. The following feed should be selected

Note: The original issue only repros after the feed from your last refresh is older than 1 hour. To simulate having an old feed run localStorage.removeItem('feedV2') in the console just before you close the browser. Depending on your platform you might also need sessionStorage.removeItem('feedV2')

[simonhong]

++
Q) Maybe this happens only when there is no feed?

[brave-builds]

A Storybook has been deployed to preview UI for the latest push

[kjozwiak]

Verification PASSED on Win 11 x64 using the following build(s):

Brave | 1.65.67 Chromium: 122.0.6261.111 (Official Build) nightly (64-bit)
-- | --
Revision | b9336aa9d2cb7900833a664f6dc9b9f25105c3bb
OS | Windows 11 Version 23H2 (Build 22631.3235)

Using the STR/Cases outlined via #22362 (comment), ensured that the last selected Feed view is saved/respected when re-opening NTP. Went through the following:

Test Case #1 - Following

• opened a NTP and selected Following as the default view
• closed the NTP/restarted Brave
• re-opened a NTP and scrolled down to the Brave News Feed and ensured Following was selected/displayed

Test Case #2 - Top Sources

• opened a NTP and selected Top Sources as the default view
• closed the NTP/restarted Brave
• re-opened a NTP and scrolled down to the Brave News Feed and ensured Top Sources was selected/displayed

Test Case #3 - Imported RSS feed

• opened a NTP and selected a Imported RSS as the default view
• closed the NTP/restarted Brave
• re-opened a NTP and scrolled down to the Brave News Feed and ensured Imported RSS was selected/displayed

Test Case #4 - For You

• opened a NTP and selected a Imported RSS as the default view
• closed the NTP/restarted Brave
• re-opened a NTP and scrolled down to the Brave News Feed and ensured Imported RSS was selected/displayed